Fatou components of elliptic polynomial skew products

TL;DR

This paper characterizes Fatou components near elliptic invariant fibers in polynomial skew-products, showing they correspond to the fiber's Fatou components under specific conditions, and explores the role of the Brjuno condition.

Contribution

It provides a detailed description of Fatou components near elliptic fibers, extending understanding of complex dynamics in two variables under certain arithmetic and geometric conditions.

Findings

Fatou components near elliptic fibers correspond to those of the fiber restriction.

The correspondence holds if the fiber's multiplier satisfies the Brjuno condition and the polynomial has no critical points on the Julia set.

The description fails when the Brjuno condition is not met.

Abstract

We investigate the description of Fatou components for polynomial skew-products in two complex variables. The non-existence of wandering domains near a super-attracting invariant fiber was shown in [L], and the geometrically-attracting case was studied in [PV] and [PS]. In [ABDPR] it was proven that wandering domains can exist near a parabolic invariant fiber. In this paper we study the remaining case, namely the dynamics near an elliptic invariant fiber. We prove that the two-dimensional Fatou components near the elliptic invariant fiber correspond exactly to the Fatou components of the restriction to the fiber, under the assumption that the multiplier at the elliptic invariant fiber satisfies the Brjuno condition and that the restriction polynomial has no critical points on the Julia set. We also show the description does not hold when the Brjuno condition is dropped. Our main tool is the construction of expanding metrics on nearby fibers, and one of the key steps in this construction is given by a local description of the dynamics near a parabolic periodic cycle.